Risk minimizing portfolios and HJBI equations for stochastic differential games

Abstract

In this paper we consider the problem to find a market portfolio that minimizes the convex risk measure of the terminal wealth in a jump diffusion market. We formulate the problem as a two player (zero-sum) stochastic differential game. To help us find a solution, we prove a theorem giving the HJBI conditions for a general zero-sum stochastic differential game in a jump diffusion setting. We then use the theorem to study particular risk minimization problems. Finally, we extend our approach to cover general stochastic differential games (not necessarily zero-sum), and we obtain similar HJBI equations for the Nash equilibria of such games.

Optimal stopping and stochastic control differential games for jump diffusions

“…Let the state, Y (t) = Y u (t), be given by equation (3), the performance functional by equation (7) and the value function by equation (8). In the following we will assume that the functions b, σ, γ, f, g are continuous with respect to (y, u).…”

“…We refer to [9] for information about optimal stopping and stochastic control for jump diffusions. The following presentation follows [8] closely.…”

Robust portfolio selection under exponential preferences

“…Maximization of the robust functional was considered, among others, by Fölmer and Gundel [3], Gundel [5], Hernández and Schied [6], Korn and Menkens [7], Korn and Wilmott [8], Mataramvura and Øksendal [9], Øk-sendal and Sulem [12], [11], Schied [15], Schied and Wu [16], and Talay and Zheng [18]. Some of these papers are based on duality arguments.…”

A maximum principle for stochastic differential games with g-expectations and partial information

“…where g i : R×R×L 2 (ν) → R are given convex functions satisfying (6). Thus E gi represents the preference of player i, i = 1, 2.…”

“…[2], [6], [9] and references therein. In this paper we study this game in the case when the performance criterion J(u) in (4) is replaced by a criterion involving risk.…”